Topological invariance of the state-integral invariants

Let $M$ be a hyperbolic $3$-manifold with a regular triangulation $\mathcal{T}$ and Neumann\Zagier datum, including a choice of quad type with $\det \mathbf{B}\neq 0$. Let $\{S_{\mathcal{T},n}\}_{n=0}^{\infty}$ be the invariants obtained from the asymptotic expansion of the associated state integral. The state integral is defined up to multiplicative factors

$$\exp\left(\frac{\pi^2}{6\hbar}a+\frac{i\pi}{4}b+\frac{\hbar}{24}c\right),\qquad a,b,c\in\mathbb{Z},$$

which induces the corresponding ambiguity in the invariants. Topological invariance conjecture. The invariants $\{S_{\mathcal{T},n}\}_{n=0}^{\infty}$ are independent of the choice of regular triangulation and Neumann\Zagier datum, up to this ambiguity, and therefore constitute topological invariants of $M$. The claim concerns the triangulation-independence of the perturbative state-integral invariants; the preceding discussion establishes their construction and describes the allowed prefactor ambiguity, but the asserted independence is presented as an expectation rather than proved here.